1. WMO International Cloud Modeling Workshop
Simulation of Hail using a Triple-moment Microphysics Scheme
Jason Milbrandt
and
M. K. Yau
WMO International Cloud Modeling Workshop
July 12, 2004

2. OBJECTIVES OF PRESENTATION:
1. Discuss the role of the shape parameter in bulk microphysics schemes
2. Demonstrate that hail sizes can be simulated using a bulk scheme
3. Compare sensitivity experiments using various bulk methods

3. Representation of a hydrometeor size distribution:
BULK METHOD
Representation of a hydrometeor size distribution:
N(D)
D [ m]
100
[m-3  m-1] 20 40 60 80
101
10-1
10-2
ANALYTIC
FUNCTION
1 m3
(unit volume)
[e.g. Cloud droplets]

4. Representation of a hydrometeor size distribution:
BULK METHOD
Representation of a hydrometeor size distribution:
GAMMA DISTRIBUTION FUNCTION
or,
etc.

5. Representation of a hydrometeor size distribution:
BULK METHOD
Representation of a hydrometeor size distribution:
GAMMA DISTRIBUTION FUNCTION
Size Distribution Parameters:
Various quantities:
N0x : “intercept”
lx : “slope”
ax : shape parameter
Qx : Mass content (Qx = r qx)
NTx: Total number concentration
Zx : Radar reflectivity
Dmx: Mean-mass diameter

6. Typical double-moment method:
BULK METHOD
Typical double-moment method:
Predict changes to Qx and NTx
Implies changes to values of the N0x and lx
(ax is held constant)

7. Alternative bulk methods:
1. Diagnostic-ax (double-moment)
- ax = f (Qx, NTx)
2. Prognostic-ax (triple-moment)
- a third moment must be predicted
e.g. add dZx/dt equation

8. How important is the shape parameter (x )
ROLE OF ALPHA
How important is the shape parameter (x )
in a bulk scheme?
Continuity equation for x:
Only SEDIMENTATION and SOURCE terms are
affected by the microphysics scheme (and hence by ax)
x: Predictive variable for hydrometeor category x

9. Analytic bin model calculation: (1D column)
ROLE OF ALPHA: 1. SEDIMENTATION
Analytic bin model calculation: (1D column) 10 5 z
(km)
Mass [g m-3] 1
1. Prescribe Q(z): D
log N(D) N0 Di
2. Compute N(Di, z):
[from a prescribed distribution]
For every size bin i:
zi(t) = zi(0) - Vi(Di)  t
3. Compute locations of each particle after sedimentation for time t:

10. Analytic bin model calculation: (1D column)
ROLE OF ALPHA: 1. SEDIMENTATION
Analytic bin model calculation: (1D column)
Mass
Content
Total Number
Concentration
Equivalent
Reflectivity
Mean-mass
Diameter
INITIAL
5 min z
[km]
10 min
15 min
20 min
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]
Contours every 5 min

11. ROLE OF ALPHA: 1. SEDIMENTATION BULK SCHEMES
= number-weighted fall velocity DM
= reflectivity-weighted fall velocity TM SM
= mass-weighted fall velocity

12. Mass Content ROLE OF ALPHA: 1. SEDIMENTATION BULK vs. ANALYTICz
[km]
Q [g m-3]
5 min
10 min
15 min
20 min
INITIAL
Analytic model:
Mass
Content
Bulk schemes:
Q [g m-3] z
[km]
DOUBLE-
MOMENT
Fixed a = 0
SINGLE-
Diagnosed a
TRIPLE-

13. ROLE OF ALPHA: 1. SEDIMENTATION BULK vs. ANALYTIC
For the prediction of NT, Z, and Dm, the various bulk methods exhibit similar relative abilities for pure sedimentation (as for Q).
NOTE:
No size-sorting mechanism exists for single-moment schemes
For the double-moment scheme with a = 0, excessive size-sorting results in very large Dm and Z

14. CONTINUOUS COLLECTION
ROLE OF ALPHA: 2. GROWTH RATES
2. MICROPHYSICS SOURCES/SINKS
CONTINUOUS COLLECTION
OF CLOUD WATER (CLcx): ( ) ( )

15. ROLE OF ALPHA: 2. GROWTH RATES
A scheme’s ability to predict the growth rates depends on its ability to compute the value of certain moments [ranging from Mx(bx) to Mx(2+bx)]
e.g. The accretion rate for hail (CLch) is proportional to Mh(2.6)
HAIL
bx  0.6

16. Analytic bin model calculation for sedimentation:
ROLE OF ALPHA: 2. GROWTH RATES
RECALL:
Analytic bin model calculation for sedimentation:
Mass
Content
Total Number
Concentration
Reflectivity z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ] Q NT Z } N0 l a { 
At each level:
M (2.6)_BULK

17. ROLE OF ALPHA: 2. GROWTH RATES (e.g. CLch)
Ratio of Growth Rates:
CLch_BULK
CLch_ ANAL
Mh(2.6)_BULK
Mh(2.6)_ANAL =
2 min 10 5
0.8
1.0
1.2
8 min 10 5
0.8
1.0
1.2 z
(km) z
(km)
Mh(2.6)_BULK
Mh(2.6)_ANAL
ANALYTIC SM
DM_FIX_0
DM_FIX_3
DM_DIAG TM
Mh(2.6)_BULK
Mh(2.6)_ANAL

18. * Milbrandt and Yau, 2004 [J. Atmos. Sci., accepted]
TESTING IN 3D: The New* Microphysics Scheme
Six hydrometeor categories:
2 liquid: cloud and rain
4 frozen: ice, snow, graupel and hail
~50 distinct microphysical processes
warm-rain scheme based on Cohard and Pinty (2000a)
ice-phase based on Murakami (1990), Ferrier (1994), Meyers et al. (1997), Reisner et al. (1998), etc.
diagnosed-a relations added for double-moment version*
predictive equations for Zx added for triple-moment version*
* Milbrandt and Yau, 2004 [J. Atmos. Sci., accepted]

19. Canadian MC2 mesoscale model - non-hydrostatic, fully compressible
CONTROL SIMULATION
MODEL:
Canadian MC2 mesoscale model
- non-hydrostatic, fully compressible
- interfaced with new microphysics scheme
(triple-moment version for CONTROL run)
CASE:
14 July 2000 “Pine Lake storm”, Alberta, Canada
- long-lasting supercell
- F3 tornado
- golf ball-sized hail

20. CONTROL SIMULATION: Nesting Strategy
12-km DOMAIN
NOTE: No CPS, perturbation, nudging (or anything else) was used to initiate the convection
ALBERTA
3-km DOMAIN
1-km DOMAIN
UTC
3 km
12 km
14 JULY
15 JULY
Model Nesting Times
1 km

21. CONTROL SIMULATION: Accumulated Total Precipitationmm 40 30 25 20 16 13 10 8 6 4
RADAR:
Accumulated Precipitation N
8:00 pm
50 km
RADAR
33 mm
1-km CNTR
8:00 pm
1-km SIMULATION:
Accumulated TOTAL Precipitation
50 km N 30 25 20 15 10 5 mm

22. CONTROL SIMULATION: Hail Swath27 26 25 24 23 22 21 20 19 18 17 16
kg m-2
VIL  27 kg m-2  LARGE HAIL
RADAR:
Composite of Maximum VIL N
8:00 pm
50 km
RADAR
1-km SIMULATION:
Accumulated SOLID Precipitation
10 mm
8:00 pm
1-km CNTR
50 km N 10 8 6 4 2 mm

23. CONTROL SIMULATION: Storm Structure: REFLECTIVITY
RADAR:
0030 UTC [6:30 pm]
1-km SIMULATION:
4:30 h [6:30 pm]
40 km
16 km
40 km
16 km
dBZ
dBZ 65 60 57 54 51 48 45 42 39 36 33 30
Maximum: 60 – 65 dBZ
COMPOSITE
Maximum: dBZ
750 hPa N N

24. CONTROL SIMULATION: Storm Structure: HOOK ECHO
RADAR:
0030 UTC [6:30 pm]
1-km SIMULATION:
4:15 h [6:15 pm]
Reflectivity CAPPI (2 km)
10 km
Equivalent Reflectivity (750 hPa)
10 km

25. CONTROL SIMULATION: Hail Sizes
How can the maximum hail sizes at the ground be inferred?
D *
LARGE
HAIL
log Nh(D)
These distributions have identical mean diameters (Dm) D
Flux of large of hail (D > D*):

26. D* = 2 cm OBSERVABLE D* = 3 cm NEGLIGIBLE
CONTROL SIMULATION: Simulated Hail Sizes
At 5:45 pm (simulation time: 4:45 h):
D* = 2 cm
Rh*(2 cm) = 5.010-2 m-2 s-1
or,
1 hailstone D  2 cm per 20 m2 every 20 seconds
OBSERVABLE
D* = 3 cm
Rh*(3 cm) = 2.310-4 m-2 s-1
or,
1.4 hailstones D  3 cm per 100 m2 every 1 minute
NEGLIGIBLE
Walnut-sized (2 – 3 cm) hail was simulated
Golf ball-sized (3 – 4 cm) hail was observed
MAXIMUM:

27. ALL RUNS USE DIFFERENT VERSIONS OF THE SAME SCHEME
SENSITIVITY EXPERIMENTS
List of Runs:
1. TRIPLE-MOMENT (control run)
2. DOUBLE-MOMENT with DIAGNOSED-a
3. DOUBLE-MOMENT with FIXED-a
(2 for r ; 0 for c, i, s, g, h)
4. SINGLE-MOMENT (similar parameters as Lin et al. 1983)
ALL RUNS USE DIFFERENT VERSIONS OF THE SAME SCHEME

28. 6-h ACCUMLATED TOTAL PRECIPITATION [mm]
SENSITIVITY EXPERIMENTS: TOTAL Precipitation
6-h ACCUMLATED TOTAL PRECIPITATION [mm]
TRIPLE-MOMENT
DOUBLE-MOMENT
Diagnosed a 33 34
SINGLE-MOMENT
DOUBLE-MOMENT
Fixed a 43 28 42
CONTOURS: 5, 10, 20, 30, 40 mm

29. 6-h ACCUMLATED SOLID PRECIPITATION [mm]
SENSITIVITY EXPERIMENTS: SOLID Precipitation (HAIL)
6-h ACCUMLATED SOLID PRECIPITATION [mm]
TRIPLE-MOMENT
DOUBLE-MOMENT
Diagnosed a 10 9 9
SINGLE-MOMENT
DOUBLE-MOMENT
Fixed a 35 25 14 23 34 13
CONTOUR INTERVAL: 2 mm

30. 700 hPa: SENSITIVITY EXPERIMENTS: Equivalent Hail Reflectivity
Zeh [dBZ]
700 hPa:
Local time: 6:30 pm
(Simulation time: 4:30 h)
TRIPLE-MOMENT
DOUBLE-MOMENT
Diagnosed a
100 km
SINGLE-MOMENT
DOUBLE-MOMENT
Fixed a

31. SENSITIVITY EXPERIMENTS: Equivalent Hail Reflectivity,
Zeh [dBZ]
Local time: 6:30 pm
(Simulation time: 4:30 h)
TRIPLE-MOMENT
63.6 dBZ
DOUBLE-MOMENT
Diagnosed a
63.6 dBZ
SINGLE-MOMENT
68.3 dBZ
DOUBLE-MOMENT
Fixed a
83.9 dBZ
10 km
0 km
25 km
50 km
75 km
100 km
MAXIMUM VALUE

32. SENSITIVITY EXPERIMENTS: Hail Mass Content,
Qh [g m-3]
Local time: 6:30 pm
(Simulation time: 4:30 h)
TRIPLE-MOMENT
5.51 g m-3
DOUBLE-MOMENT
Diagnosed a
5.58 g m-3
SINGLE-MOMENT
3.71 g m-3
DOUBLE-MOMENT
Fixed a
4.91 g m-3
Dashed contour: 0.1 g m-3
MAXIMUM VALUE

33. SENSITIVITY EXPERIMENTS: Hail Number Concentration
log NTh [m-3]
Local time: 6:30 pm
(Simulation time: 4:30 h)
TRIPLE-MOMENT
5.18
DOUBLE-MOMENT
Diagnosed a
4.07
SINGLE-MOMENT
1.53
DOUBLE-MOMENT
Fixed a
5.22
Dashed contour: 1.0 m-3
MAXIMUM VALUE

34. SENSITIVITY EXPERIMENTS: Mean Hail Diameters,
Dmh [mm]
Local time: 6:30 pm
(Simulation time: 4:30 h)
TRIPLE-MOMENT
14.9 mm
DOUBLE-MOMENT
Diagnosed a
11.2 mm
SINGLE-MOMENT
6.15 mm
DOUBLE-MOMENT
Fixed a
67.2 mm
MAXIMUM VALUE

35. SENSITIVITY EXPERIMENTS: Large Hail Concentration,
Nh*{1 cm} [m-3] (grape-sized or larger)
Local time: 6:30 pm
(Simulation time: 4:30 h)
TRIPLE-MOMENT
1.03 m-3
DOUBLE-MOMENT
Diagnosed a
0.79 m-3
SINGLE-MOMENT
1.76 m-3
DOUBLE-MOMENT
Fixed a
0.68 m-3
Dashed contour: m-3
MAXIMUM VALUE

36. SENSITIVITY EXPERIMENTS: Maximum hail sizes (at surface)
3 – 4 cm (Golf ball-sized ) hail was observed
[at 5:45 pm, time of maximum hail rate in CONTROL RUN]
TRIPLE-MOMENT
DOUBLE-MOMENT
Diagnosed a
2 – 3 cm (Walnut-sized)
1 – 2 cm (Grape-sized)
SINGLE-MOMENT
DOUBLE-MOMENT
Fixed a
4 – 5 cm (Baseball-sized)
8 – 9 cm (Grapefruit-sized)

37. CONCLUSIONS    THANK YOU
1. The value of the shape parameter is important in bulk microphysics schemes
TRIPLE-
MOMENT
DOUBLE-
Diagnosed a
SINGLE-
Fixed a   
2. For the overall QPF, storm structure, hydrometeor values, and the simulation of hail sizes:
THANK YOU

39. SINGLE-moment bulk scheme (SM):z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]
ANALYTIC BIN model (ANA): z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]

40. DOUBLE-moment bulk scheme (FIX0): a = 0z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]
ANALYTIC BIN model (ANA): z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]

41. DOUBLE-moment bulk scheme (FIX3): a = 3z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]
ANALYTIC BIN model (ANA): z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]

42. TRIPLE-moment bulk scheme (TM):z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]
ANALYTIC BIN model (ANA): z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]

43. DOUBLE-moment bulk scheme (DIAG): a = f(Dm)z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]
ANALYTIC BIN model (ANA): z
[km]
Q [g m-3]
NT [m-3]
Ze [dBZ]
Dm [mm]